Question 1

To answer this question, please start by builiding and calibrating a 10-period Black-Derman-Toy model for the short-rate, ri,j. You may assume that the term-structure of interest rates observed in the market place is:

Period 1 2 3 4 5 6 7 8 9 10

Spot Rate 3.0% 3.1% 3.2% 3.3% 3.4% 3.5% 3.55% 3.6% 3.65% 3.7%

As in the video modules, these interest rates assume per-period compounding. For example, the market-price of a zero-coupon bond that matures in period 6 is Z_0^6 = 100/(1+.035)^6 = 81.35 assuming a face value of 100.

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Assume b=0.05 is a constant for all ii in the BDT model as we assumed in the video lectures. Calibrate the a_iai parameters so that the model term-structure matches the market term-structure. Be sure that the final error returned by Solver is at most 10^{-8} (This can be achieved by rerunning Solver multiple times if necessary, starting each time with the solution from the previous call to Solver.)

Once your model has been calibrated, compute the price of a payer swaption with notional $1M that expires at time t=3 with an option strike of 0. You may assume the underlying swap has a fixed rate of 3.9% and that if the option is exercised then cash-flows take place at times t=4,,10. (The cash-flow at time t=it=i is based on the short-rate that prevailed in the previous period, i.e. the payments of the underlying swap are made in arrears.)

Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be 10,456.67, submit 10457.